50 ANALYTICAL MECHANICS words, (£,y) is the point of application of the resultant fore*. The resultant force is evidently parallel to the given iom*. The last two equations may be written m the following (XIII) JR m JB ILLUSTRATIVE EXAMPLE. Find the resultant of two parallel forces which act upon a rigid body in the same direction. Let the 2/-axis be parallel to the forces. Then R - and x - i + j or 9- =-------- Fz x — xi But since xz — x and x — x\ are the distances of F2 and Fi from R, we have * i Fiu. 37. or Fidi = W2. Therefore the distances of the resultant from the given forces are inversely proportional to the magnitudes of the latter. PROBLEMS. 1. Find the resultant force and the resultant torque duo to the forces P}2P,4P and 2 P which act along the sides of a square, taken in order. 2. Three forces are represented in magnitude and line of action by the sides of an equilateral triangle. Find the resultant force, taking tlw directions of one of the forces opposite to that of the other two. 3. The lines of action of three forces form a right LsoneeleH triangle of sides a, a, and a V2. The magnitudes of the forces are proportional to the sides of the triangle. Find the resultant force. 4. The sum of the moments of a system of coplanar forces about any three points, which are not in the same straight line, are the same. Show that the system is equivalent to a couple.